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Math Help - Connected Space

  1. #1
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    Connected Space

    Hello, Can anyone give some clue on how I can determine if R^{Z^+} is or isn't connected in the metric topology with distance
    \rho(x,y)=Sup\{\overline{d}(x_i,y_i):i\in Z^+\} where
    \overline{d}(a,b)=Min\{|a-b|,1\}
    Last edited by facenian; October 21st 2010 at 02:23 PM. Reason: gramma fixing
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  2. #2
    Senior Member Tinyboss's Avatar
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    Look at the bounded and unbounded sequences.
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  3. #3
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    I think all sequenses are bounded because \rho(x,y)\leq 1 for all x,y
    Last edited by facenian; October 24th 2010 at 04:57 AM. Reason: gramma fixing
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  4. #4
    Senior Member Tinyboss's Avatar
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    Not bounded in that sense. The space \mathbb{R}^{\mathbb{Z}^+} is the space of sequences of real numbers, and such a sequence is bounded iff there's some real number M larger than the absolute value of every element of the sequence. Show that in the given topology, the set of bounded (respectively unbounded) sequences is open.
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